Contents

Idea

Recall that a category consists of a collection of morphisms each having a single object as source or input, and a single object as target or output, together with laws for composition and identity obeying associativity and identity axioms. A multicategory is like a category, except that one allows multiple inputs and a single output.

Many people (especially non-category theorists) use the word multicategory or the word colored operad to mean what we would call a symmetric multicategory / symmetric operad. These are multicategories equipped with an action of the symmetric groupSnS_n on the multimorphisms c1,…,cn→cc_1, \ldots, c_n \to c such that the composition is equivariant with respect to these actions.

In terms of cartesian monads

An efficient abstract method for defining multicategories and related structures is through the formalism of cartesian monads. For ordinary categories, one uses the identity monad on Set; for ordinary multicategories, one uses the free monoid monad (−)*:Set→Set(-)*: Set \to Set. This is a special case of the yet more general notion of generalized multicategory.

First, a TT-span from XX to YY is a spanpp from TXT X to YY, that is, a diagram

TX←p1P→p2YT X \stackrel{p_1}{\leftarrow} P \stackrel{p_2}{\to} Y

A TT-span is often written as p:X⇸Yp: X &#x21F8; Y.

When TT is the free monoid monad on SetSet, a TT-span from XX to itself is called a multigraph on XX.

TT-spans are the 1-cells of a bicategory. A 2-cell between TT-spans e,f:X⇸Ye, f: X &#x21F8; Y is a 2-cell between ordinary spans from TXT X to YY. To horizontally compose TT-spans e:X⇸Ye: X &#x21F8; Y and f:Y⇸Zf: Y &#x21F8; Z, take the ordinary span composite of